Lower bounds for the weak-type constants of the operators $\Lambda_m$

TL;DR

This paper establishes lower bounds for the weak-type (1,1) constants of operators Lambda_m related to the Beurling-Ahlfors transform, disproving a previous conjecture and analyzing their asymptotic behavior.

Contribution

It provides the first known lower bounds for Lambda_m constants and shows they do not tend to 1 as m increases, challenging prior assumptions.

Findings

Weak-type (1,1) constant of Lambda_0 is approximately 1.44.

Lower bounds for Lambda_1 exceed 1.38.

Constants for Lambda_m do not approach 1 as m  infinity.

Abstract

The operators $\Lambda_m$ ($m\in\mathbb{N}\cup \{0\}$) arise when one studies the action of the Beurling-Ahlfors transform on certain radial function subspaces. It is known that the weak-type $(1,1)$ constant of $\Lambda_0$ is equal to $1/\ln(2)\approx 1.44$. We construct examples showing that the weak-type $(1,1)$ constant of $\Lambda_1$ is larger than $1.38$ and that the weak-type $(1,1)$ constant of $\Lambda_m$ does not tend to $1$ when $m\to\infty$. This disproves a conjecture of Gill [Mich. Math. J. 59 (2010), No. 2, 353-363]. We also prove a companion result for the adjoint operators. This is the arXiv version of the paper - it includes some additional discussion in the appendices.